Conventionally, a calibration device that performs camera calibration of an image-acquisition device or a projection device is known (for example, see PTL 1). A camera model includes a plurality of unknown parameters (camera parameters). By obtaining the camera parameters with the calibration device, it is possible to mathematically obtain backprojection lines in the real world corresponding to two-dimensional coordinates in an image.
Here, the conventional camera calibration, disclosed in PTL 1 and NTL 1, will be described. The camera calibration is performed in the following procedure by using a mathematical camera model that expresses a process in which three-dimensional coordinates in the real world are image-captured by a camera and are converted into two-dimensional coordinates in an image. First, using Expression 1, three-dimensional coordinates in the real world (hereinbelow, world coordinates) (x, y, z) are projected on normalized image plane coordinates (up, vp).
                    {                                                                              u                  p                                =                                                                                                    r                        11                                            ⁢                      x                                        +                                                                  r                        12                                            ⁢                      y                                        +                                                                  r                        13                                            ⁢                      z                                        +                                          t                      x                                                                                                                          r                        31                                            ⁢                      x                                        +                                                                  r                        32                                            ⁢                      y                                        +                                                                  r                        33                                            ⁢                      z                                        +                                          t                      z                                                                                                                                                                v                  p                                =                                                                                                    r                        21                                            ⁢                      x                                        +                                                                  r                        22                                            ⁢                      y                                        +                                                                  r                        23                                            ⁢                      z                                        +                                          t                      y                                                                                                                          r                        31                                            ⁢                      x                                        +                                                                  r                        32                                            ⁢                      y                                        +                                                                  r                        33                                            ⁢                      z                                        +                                          t                      z                                                                                                                              {                  Expression          ⁢                                          ⁢          1                }                                          R          =                      (                                                                                r                    11                                                                                        r                    12                                                                                        r                    13                                                                                                                    r                    21                                                                                        r                    22                                                                                        r                    23                                                                                                                    r                    31                                                                                        r                    32                                                                                        r                    33                                                                        )                          ,                  T          =                      (                                                                                t                    x                                                                                                                    t                    y                                                                                                                    t                    z                                                                        )                                              {                  Expression          ⁢                                          ⁢          2                }            
Note that the rotation matrix R and the translation vector T in Expression 2 express three-dimensional coordinate conversion from the world coordinates to the camera coordinates. These are the values showing the position and orientation of the camera with respect to the world coordinates and are called “extrinsic parameters”.
Note that Expression 1 is based on an assumption that all backprojection lines intersect at the optical center of the camera. Next, using Expression 3, coordinates (ud, vd) obtained by adding distortion to the normalized image plane coordinates (up, vp) are obtained.
                    {                                                                              u                  d                                =                                                      u                    p                                    +                                                            g                      1                                        ⁡                                          (                                                                        u                          p                          2                                                +                                                  v                          p                          2                                                                    )                                                        +                                                            g                      3                                        ⁢                                          u                      p                      2                                                        +                                                            g                      4                                        ⁢                                          u                      p                                        ⁢                                          v                      p                                                        +                                                            k                      1                                        ⁢                                                                  u                        p                                            ⁡                                              (                                                                              u                            p                            2                                                    +                                                      v                            p                            2                                                                          )                                                                                                                                                                                      v                  d                                =                                                      v                    p                                    +                                                            g                      2                                        ⁡                                          (                                                                        u                          p                          2                                                +                                                  v                          p                          2                                                                    )                                                        +                                                            g                      3                                        ⁢                                          u                      p                                        ⁢                                          v                      p                                                        +                                                            g                      4                                        ⁢                                          v                      p                      2                                                        +                                                            k                      1                                        ⁢                                                                  v                        p                                            ⁡                                              (                                                                              u                            p                            2                                                    +                                                      v                            p                            2                                                                          )                                                                                                                                                    {                  Expression          ⁢                                          ⁢          3                }            
Note that (g1, g2, g3, g4, k1) are distortion parameters. Furthermore, using Expression 4, the normalized image plane coordinates (ud, vd) obtained by adding the distortion are converted into pixel-unit-based pixel coordinates (u, v).
                    {                                                            u                =                                                                            α                      u                                        ⁢                                          u                      d                                                        +                                      u                    0                                                                                                                          v                =                                                                            α                      v                                        ⁢                                          v                      d                                                        +                                      v                    0                                                                                                          {                  Expression          ⁢                                          ⁢          4                }            
In a standard camera model, conversion from the world coordinates (x, y, z), obtained by image-acquisition with the camera, into the pixel coordinates (u, v) is expressed with Expressions 1 to 4 in this way. Because parameters (αu, αv, u0, v0, g1, g2, g3, g4, k1) in Expression 3 and Expression 4 represent the properties of the camera itself, they are called “intrinsic parameters”.
The distortion parameters are variously defined according to the usage. For example, although Expression 3 expresses a model in which distortion of up to third order is taken into consideration, a model in which a term of a higher order, such as a fifth, a seventh, or a higher order, is added, is also used. Among them, a representative distortion model is Brown's model disclosed in NPL 2, shown in Expression 5.
                              (                                                                      u                  d                                                                                                      v                  d                                                              )                =                              (                                                                                u                    p                                                                                                                    v                    p                                                                        )                    +                                    (                                                                    k                    1                                    ⁢                                      r                    p                    2                                                  +                                                      k                    2                                    ⁢                                      r                    p                    4                                                  +                                                      k                    3                                    ⁢                                      r                    p                    6                                                  +                …                            ⁢                                                          )                        ⁢                          (                                                                                          u                      p                                                                                                                                  v                      p                                                                                  )                                +                                    [                                                          ⁢                                                                    p                    1                                    ⁡                                      (                                                                                                                                                      r                              p                              2                                                        +                                                          2                              ⁢                                                                                                                          ⁢                                                              u                                p                                2                                                                                                                                                                                                                                      2                            ⁢                                                                                                                  ⁢                                                          u                              p                                                        ⁢                                                          v                              p                                                                                                                                            )                                                  +                                                      p                    2                                    ⁡                                      (                                                                                                                        2                            ⁢                                                                                                                  ⁢                                                          u                              p                                                        ⁢                                                          v                              p                                                                                                                                                                                                                                      r                              p                              2                                                        +                                                          2                              ⁢                                                                                                                          ⁢                                                              v                                p                                2                                                                                                                                                                          )                                                              ]                        ⁢                          (                              1                +                                                      p                    3                                    ⁢                                      r                    p                    2                                                  +                …                            ⁢                                                          )                                                          {                  Expression          ⁢                                          ⁢          5                }            where rp2=up2+vp2 
In Brown's model, distortion is represented by parameters (k1, k2, k3, . . . ) of rotationally symmetrical radial distortion and parameters (p1, p2, p3, . . . ) of rotationally asymmetrical tangential distortion.
Typically, in camera calibration, an image of a calibration chart having a plurality of feature points whose world coordinates (x, y, z) are known is captured with a camera. Subsequently, through image processing, the pixel coordinates (u, v) at which the feature points are image-captured are acquired. In this way, a plurality of measurement data representing the correspondence between the world coordinates (x, y, z) and the pixel coordinates (u, v) are obtained, thereby obtaining the camera parameters.